diff --git a/content/blog/1d-collisions/index.md b/content/blog/1d-collisions/index.md index 106d539..2601f64 100644 --- a/content/blog/1d-collisions/index.md +++ b/content/blog/1d-collisions/index.md @@ -1,29 +1,51 @@ -$$ m_{1}\vec{v}_{1} + m_{2}\vec{v}_{2} = m_{1}\vec{v'}_{1} + m_{2}\vec{v'}_{2} $$ +$$ +m_{1}\vec{v}_{1} + m_{2}\vec{v}_{2} = m_{1}\vec{v'}_{1} + m_{2}\vec{v'}_{2} +$$ -$$ \frac{m_{1}{v_{1}}^2}{2} + \frac{m_{2}{v_{2}}^2}{2} = \frac{m_{1}{v'_{1}}^2}{2} + \frac{m_{2}{v'_{2}}^2}{2} $$ +$$ +\frac{m_{1}{v_{1}}^2}{2} + \frac{m_{2}{v_{2}}^2}{2} = \frac{m_{1}{v'_{1}}^2}{2} + \frac{m_{2}{v'_{2}}^2}{2} +$$ -$$\begin{cases}m_{1}({v_{1}} - {v'_{1}}) = m_{2}({v'_{2}} - {v_{2}})\\m_{1}({v_{1}}^2 - {v'_{1}}^2) = m_{2}({v'_{2}}^2 - {v_{2}}^2) -\end{cases}$$ +$$ +\begin{cases} +m_{1}({v_{1}} - {v'_{1}}) = m_{2}({v'_{2}} - {v_{2}}) \\ +m_{1}({v_{1}}^2 - {v'_{1}}^2) = m_{2}({v'_{2}}^2 - {v_{2}}^2) +\end{cases} +$$ -$$ \frac{v_{1} - v'_{1}}{{v_{1}^2} - {v'_{1}}^2} = \frac{v'_{2} - v_{2}}{{{v'_{2}}^2} - {v_{2}}^2} $$ +$$ +\frac{v_{1} - v'_{1}}{{v_{1}^2} - {v'_{1}}^2} = \frac{v'_{2} - v_{2}}{{{v'_{2}}^2} - {v_{2}}^2} +$$ -$$ \frac{v_{1} - v'_{1}}{(v_1 + v'_1)(v_1 - v'_1)} = \frac{v'_{2} - v_{2}}{(v'_2 - v_2)(v'_2 + v_2)} $$ +$$ +\frac{v_{1} - v'_{1}}{(v_1 + v'_1)(v_1 - v'_1)} = \frac{v'_{2} - v_{2}}{(v'_2 - v_2)(v'_2 + v_2)} +$$ -$$ \frac{1}{v_1 + v'_1} = \frac{1}{v'_2 + v_2} $$ +$$ +\frac{1}{v_1 + v'_1} = \frac{1}{v'_2 + v_2} +$$ -$$ v_1 + v'_1 = v'_2 + v_2 $$ +$$ +v_1 + v'_1 = v'_2 + v_2 +$$ -$$\begin{cases} -v'_1 = v_2 + v'_2 - v_1\\ +$$ +\begin{cases} +v'_1 = v_2 + v'_2 - v_1 \\ v'_2 = v_1 + v'_1 - v_2 -\end{cases}$$ +\end{cases} +$$ -$$ \begin{cases} -m_1v_1 + m_2v_2 = m_1{v'}_1 + m_2(v_1 + {v'}_1 - v_2)\\ +$$ +\begin{cases} +m_1v_1 + m_2v_2 = m_1{v'}_1 + m_2(v_1 + {v'}_1 - v_2) \\ m_1v_1 + m_2v_2 = m_1(v_2 + v'_2 - v_1) + m_2v'_2 -\end{cases} $$ +\end{cases} +$$ -$$ \begin{cases} -v'_1 = \frac{2m_2v_2 + v_1(m_1-m_2)}{m_1+m_2}\\ +$$ +\begin{cases} +v'_1 = \frac{2m_2v_2 + v_1(m_1-m_2)}{m_1+m_2} \\ v'_2 = \frac{2m_1v_1 + v_2(m_2-m_1)}{m_1+m_2} -\end{cases} $$ +\end{cases} +$$